Course: 8.11 指数函数图 | The Way To Learn

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指数函数图图

Graphs of Exponential Functions
::指数函数图图

A colony of bacteria has a population of three thousand at noon on Monday. During the next week, the colony’s population doubles every day. What is the population of the bacteria colony just before midnight on Saturday?
::星期一中午,一个细菌聚居区的人口为3,000人。下周,该聚居区的人口每天翻一番。就在星期六午夜午夜之前,细菌聚居区的人口是多少?

At first glance, this seems like a problem you could solve using a geometric sequence . And you could, if the bacteria population doubled all at once every day; since it doubled every day for five days, the final population would be 3000 times $\begin{aligned} 2^{5} \end{aligned}$ .
::乍一看,这似乎是一个你可以使用几何序列来解决的问题。而且,如果细菌数量每天一次翻一番,那么,如果细菌数量每天翻一番,5天每天翻一番,那么最终人口将达到3000倍25。

But bacteria don’t reproduce all at once; their population grows slowly over the course of an entire day. So how do we figure out the population after five and a half days?
::但细菌不会同时出现所有繁殖;它们的人口在一整天中缓慢增长。那么我们如何在五天半后找到人口呢?

Exponential Functions
::指数函数

are a lot like geometrical sequences. The main difference between them is that a geometric sequence is discrete while an exponential function is continuous .
::与几何序列非常相似。它们之间的主要区别是几何序列是离散的,而指数函数是连续的。

Discrete means that the sequence has values only at distinct points (the 1st term , 2nd term, etc.)
::分辨是指序列仅在不同点(第一个条件、第二个条件等)才有数值。

Continuous means that the function has values for all possible values of $\begin{aligned} x \end{aligned}$ . The integers are included, but also all the numbers in between.
::连续表示函数为 x 的所有可能值都有值。包含整数,但也包括中间的所有数字。

The problem with the bacteria is an example of a continuous function. Here’s an example of a discrete function:
::细菌的问题是一个连续函数的例子。这里的例子是一个离散函数:

An ant walks past several stacks of Lego blocks. There is one block in the first stack, 3 blocks in the $\begin{aligned} 2^{n d} \end{aligned}$ stack and 9 blocks in the $\begin{aligned} 3^{r d} \end{aligned}$ stack. In fact, in each successive stack there are triple the number of blocks than in the previous stack.
::蚂蚁走过 Lego 区块的几层。第一层有一个街区, 第二层有三个街区, 第三层九个街区。事实上, 在每一层, 每层的区块数是前层的三倍。

In this example, each stack has a distinct number of blocks and the next stack is made by adding a certain number of whole pieces all at once. More importantly, however, there are no values of the sequence between the stacks. You can’t ask how high the stack is between the $\begin{aligned} 2^{n d} \end{aligned}$ and $\begin{aligned} 3^{r d} \end{aligned}$ stack, as no stack exists at that position!
::在此示例中, 每个堆叠都有不同数量的区块, 下一个堆叠会同时添加一定数量的整块。但是, 更重要的是, 堆叠之间的序列没有数值。您无法询问堆叠在第二和第三堆之间的高度, 因为该位置没有堆叠 !

As a result of this difference, we use a geometric series to describe quantities that have values at discrete points, and we use exponential functions to describe quantities that have values that change continuously.
::由于这一差异,我们使用一个几何序列来描述在离散点具有数值的数量,我们使用指数函数来描述具有持续变化数值的数量。

When we graph an exponential function, we draw the graph with a solid curve to show that the function has values at any time during the day. On the other hand, when we graph a geometric sequence, we draw discrete points to signify that the sequence only has value at those points but not in between.
::当我们用固体曲线绘制指数函数时,我们绘制图形以显示该函数在白天的任何时候都有数值。另一方面,当我们绘制几何序列时,我们绘制离散点,以表示该序列在这些点上只有价值,而不是介于这些点之间。

Here are graphs for the two examples above:
::以上两个例子的图表如下:

The formula for an exponential function is similar to the formula for finding the terms in a geometric sequence. An exponential function takes the form
::指数函数的公式与在几何序列中查找术语的公式相似。指数函数以

\begin{aligned} y = A \cdot b^{x} \end{aligned}

::y=A=Bx y=A=Bx

where $\begin{aligned} A \end{aligned}$ is the starting amount and $\begin{aligned} b \end{aligned}$ is the amount by which the total is multiplied every time. For example, the bacteria problem above would have the equation $\begin{aligned} y = 3000 \cdot 2^{x} \end{aligned}$ .
::此处 A 是起始数, b 是总数每次乘以的数值。例如, 上面的细菌问题将包含 y= 3000% 2x 的等式。

Compare Graphs of Exponential Functions
::比较指数函数图

Let’s graph a few exponential functions and see what happens as we change the constants in the formula. The basic shape of the exponential function should stay the same—but it may become steeper or shallower depending on the constants we are using.
::让我们绘制几个指数函数图,看看当我们改变公式中的常数时会发生什么。指数函数的基本形状应该保持不变 — — 但根据我们使用的常数,指数函数可能会更加陡峭或浅。

First, let’s see what happens when we change the value of $\begin{aligned} A \end{aligned}$ .
::首先,让我们看看当我们改变A的价值时会发生什么。

Compare the graphs of $\begin{aligned} y = 2^{x} \end{aligned}$ and $\begin{aligned} y = 3 \cdot 2^{x} \end{aligned}$ .
::比较 y= 2x 和 y= 3 = 2x 的图形。

Let’s make a table of values for both functions.
::让我们为这两个函数绘制一个数值表。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = 2^{x} \end{aligned}$	$\begin{aligned} y = 3 \cdot 2^{x} \end{aligned}$
-3	$\begin{aligned} \frac{1}{8} \end{aligned}$	$\begin{aligned} y = 3 \cdot 2^{- 3} = 3 \cdot \frac{1}{2^{3}} = \frac{3}{8} \end{aligned}$
-2	$\begin{aligned} \frac{1}{4} \end{aligned}$	$\begin{aligned} y = 3 \cdot 2^{- 2} = 3 \cdot \frac{1}{2^{2}} = \frac{3}{4} \end{aligned}$
-1	$\begin{aligned} \frac{1}{2} \end{aligned}$	$\begin{aligned} y = 3 \cdot 2^{- 1} = 3 \cdot \frac{1}{2^{1}} = \frac{3}{2} \end{aligned}$
0	1	$\begin{aligned} y = 3 \cdot 2^{0} = 3 \end{aligned}$
1	2	$\begin{aligned} y = 3 \cdot 2^{1} = 6 \end{aligned}$
2	4	$\begin{aligned} y = 3 \cdot 2^{2} = 3 \cdot 4 = 12 \end{aligned}$
3	8	$\begin{aligned} y = 3 \cdot 2^{3} = 3 \cdot 8 = 24 \end{aligned}$

Now let's use this table to graph the functions.
::现在让我们用这个表格来绘制函数的图形。

We can see that the function $\begin{aligned} y = 3 \cdot 2^{x} \end{aligned}$ is bigger than the function $\begin{aligned} y = 2^{x} \end{aligned}$ . In both functions, the value of $\begin{aligned} y \end{aligned}$ doubles every time $\begin{aligned} x \end{aligned}$ increases by one. However, $\begin{aligned} y = 3 \cdot 2^{x} \end{aligned}$ “starts” with a value of 3, while $\begin{aligned} y = 2^{x} \end{aligned}$ “starts” with a value of 1, so it makes sense that $\begin{aligned} y = 3 \cdot 2^{x} \end{aligned}$ would be bigger as its values of $\begin{aligned} y \end{aligned}$ keep getting doubled.
::我们可以看到, y= 3 = 2x 的函数大于 y= 2x 的函数。在两个函数中, Y 的值每增加一次就会加倍。然而, y= 3 = 2x “ 启动” 值为 3, 而 y= 2x “启动” 值为 1, 因此, y= 3 = 2x 的值会随着 Y 的值不断加倍而更大是有道理的。

Similarly, if the starting value of $\begin{aligned} A \end{aligned}$ is smaller, the values of the entire function will be smaller.
::同样,如果A的起始值较小,整个功能的起始值也将较小。

Comparing Graphs
::比较图表

Compare the graphs of $\begin{aligned} y = 2^{x} \end{aligned}$ and $\begin{aligned} y = \frac{1}{3} \cdot 2^{x} \end{aligned}$ .
::比较 y= 2x 和 y= 13. 2x 的图形。

Let’s make a table of values for both functions.
::让我们为这两个函数绘制一个数值表。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = 2^{x} \end{aligned}$	$\begin{aligned} y = \frac{1}{3} \cdot 2^{x} \end{aligned}$
-3	$\begin{aligned} \frac{1}{8} \end{aligned}$	$\begin{aligned} y = \frac{1}{3} \cdot 2^{- 3} = \frac{1}{3} \cdot \frac{1}{2^{3}} = \frac{1}{24} \end{aligned}$
-2	$\begin{aligned} \frac{1}{4} \end{aligned}$	$\begin{aligned} y = \frac{1}{3} \cdot 2^{- 2} = \frac{1}{3} \cdot \frac{1}{2^{2}} = \frac{1}{12} \end{aligned}$
-1	$\begin{aligned} \frac{1}{2} \end{aligned}$	$\begin{aligned} y = \frac{1}{3} \cdot 2^{- 1} = \frac{1}{3} \cdot \frac{1}{2^{1}} = \frac{1}{6} \end{aligned}$
0	1	$\begin{aligned} y = \frac{1}{3} \cdot 2^{0} = \frac{1}{3} \end{aligned}$
1	2	$\begin{aligned} y = \frac{1}{3} \cdot 2^{1} = \frac{2}{3} \end{aligned}$
2	4	$\begin{aligned} y = \frac{1}{3} \cdot 2^{2} = \frac{1}{3} \cdot 4 = \frac{4}{3} \end{aligned}$
3	8	$\begin{aligned} y = \frac{1}{3} \cdot 2^{3} = \frac{1}{3} \cdot 8 = \frac{8}{3} \end{aligned}$

Now let's use this table to graph the functions.
::现在让我们用这个表格来绘制函数的图形。

As we expected, the exponential function $\begin{aligned} y = \frac{1}{3} \cdot 2^{x} \end{aligned}$ is smaller than the exponential function $\begin{aligned} y = 2^{x} \end{aligned}$ .
::正如我们所预期的那样, 指数函数y=132x小于指数函数y=2x。

So what happens if the starting value of $\begin{aligned} A \end{aligned}$ is negative? Let’s find out.
::如果A的起始值为负值,会怎么样?让我们来看看。

Example C
::例例C

Graph the exponential function $\begin{aligned} y = - 5 \cdot 2^{x} \end{aligned}$ .
::图形显示指数函数====================================================================================================x====================================================================================================================================================================================================================================================================================================================================================================================================================

Solution
::解决方案

Let’s make a table of values:
::让我们绘制一个数值表:

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = - 5 \cdot 2^{x} \end{aligned}$
-2	$\begin{aligned} - \frac{5}{4} \end{aligned}$
-1	$\begin{aligned} - \frac{5}{2} \end{aligned}$
0	-5
1	-10
2	-20
3	-40

Now let's graph the function:
::现在让我们绘制函数图 :

This result shouldn’t be unexpected. Since the starting value is negative and keeps doubling over time, it makes sense that the value of $\begin{aligned} y \end{aligned}$ gets farther from zero, but in a negative direction. The graph is basically just like the graph of $\begin{aligned} y = 5 \cdot 2^{x} \end{aligned}$ , only mirror-reversed about the $\begin{aligned} x - \end{aligned}$ axis.
::这一结果不应该出乎意料。由于起始值为负值,并且随着时间的推移持续翻一番, y的值从零到负方向都更远是有道理的。图表基本上和y=5=2x的图一样,只是对 x- 轴的镜反转。

Now, let’s compare exponential functions whose bases $\begin{aligned} (b) \end{aligned}$ are different.
::现在,让我们比较一下基准(b)不同的指数函数。

Graphing Multiple Functions
::绘制多个函数

Graph the following exponential functions on the same graph: $\begin{aligned} y = 2^{x}, y = 3^{x}, y = 5^{x}, y = 10^{x} \end{aligned}$ .
::在同一图中绘制下列指数函数:y=2x,y=3x,y=5x,y=10x。

First we’ll make a table of values for all four functions.
::首先,我们将为所有四个函数绘制一个数值表。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = 2^{x} \end{aligned}$	$\begin{aligned} y = 3^{x} \end{aligned}$	$\begin{aligned} y = 5^{x} \end{aligned}$	$\begin{aligned} y = 10^{x} \end{aligned}$
-2	$\begin{aligned} \frac{1}{4} \end{aligned}$	$\begin{aligned} \frac{1}{9} \end{aligned}$	$\begin{aligned} \frac{1}{25} \end{aligned}$	$\begin{aligned} \frac{1}{100} \end{aligned}$
-1	$\begin{aligned} \frac{1}{2} \end{aligned}$	$\begin{aligned} \frac{1}{3} \end{aligned}$	$\begin{aligned} \frac{1}{5} \end{aligned}$	$\begin{aligned} \frac{1}{10} \end{aligned}$
0	1	1	1	1
1	2	3	5	10
2	4	9	25	100
3	8	27	125	1000

Now let's graph the functions.
::现在让我们来绘制函数的图表。

Notice that for $\begin{aligned} x = 0 \end{aligned}$ , all four functions equal 1. They all “start out” at the same point, but the ones with higher values for $\begin{aligned} b \end{aligned}$ grow faster when $\begin{aligned} x \end{aligned}$ is positive—and also shrink faster when $\begin{aligned} x \end{aligned}$ is negative.
::注意对于 x=0 , 所有四个函数都等于 1 。它们都在同一点“ 启动 ” , 但对于 b 值较高的, 当 x 呈正数时增长更快, 当 x 呈负数时增长更快, 当 x 呈负数时收缩更快。

Finally, let’s explore what happens for values of $\begin{aligned} b \end{aligned}$ that are less than 1.
::最后,让我们探讨B值小于1的情况。

Example E
::例E

Graph the exponential function $\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{x} \end{aligned}$ .
::图形显示指数函数=5(12)x。

Solution
::解决方案

Let’s start by making a table of values. (Remember that a fraction to a negative power is equivalent to its reciprocal to the same positive power.)
::让我们从一个价值表开始。 (记住,一个负强的一小部分相当于其对同一种积极力量的对等性。 )

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{x} \end{aligned}$
-3	$\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{- 3} = 5 \cdot 2^{3} = 40 \end{aligned}$
-2	$\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{- 2} = 5 \cdot 2^{2} = 20 \end{aligned}$
-1	$\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{- 1} = 5 \cdot 2^{1} = 10 \end{aligned}$
0	$\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{0} = 5 \cdot 1 = 5 \end{aligned}$
1	$\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{1} = \frac{5}{2} \end{aligned}$
2	$\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{2} = \frac{5}{4} \end{aligned}$

Now let's graph the function.
::现在让我们来绘制函数的图表。

This graph looks very different than the graphs from the previous example! What’s going on here?
::这个图表看起来与上一个例子的图表大不相同! 这里发生了什么?

When we raise a number greater than 1 to the power of $\begin{aligned} x \end{aligned}$ , it gets bigger as $\begin{aligned} x \end{aligned}$ gets bigger. But when we raise a number smaller than 1 to the power of $\begin{aligned} x \end{aligned}$ , it gets smaller as $\begin{aligned} x \end{aligned}$ gets bigger—as you can see from the table of values above. This makes sense because multiplying any number by a quantity less than 1 always makes it smaller.
::当我们把一个数字加到x的功率上时,它会随着x的功率增加而变得更大。但是当我们把一个数字加到x的功率上时,它会随着x的功率增加而变小——从上面的数值表中可以看到这一点。这有道理,因为任何数字乘以一个小于1的数量总是使它变小。

So, when the base $\begin{aligned} b \end{aligned}$ of an exponential function is between 0 and 1, the graph is like an ordinary exponential graph, only decreasing instead of increasing. Graphs like this represent instead of . Exponential decay functions are used to describe quantities that decrease over a period of time.
::因此,当指数函数的基 b 在 0 和 1 之间时, 图形就像普通的指数图形, 仅减少而不是增加。像这样的图表代表着。指数衰减函数用来描述一段时间内减少的数量。

When $\begin{aligned} b \end{aligned}$ can be written as a fraction, we can use the Property of Negative Exponents to write the function in a different form. For instance, $\begin{aligned} y = 5 \cdot {(\frac{1}{2})}^{x} \end{aligned}$ is equivalent to $\begin{aligned} 5 \cdot 2^{- x} \end{aligned}$ . These two forms are both commonly used, so it’s important to know that they are equivalent.
::当 b 可以作为一个分数写入时, 我们可以使用负指数属性以不同的形式写入函数。例如, y= 5( 12)x 相当于 52- x。这两种形式都常用, 因此重要的是要知道它们是等值的。

Examples
::实例

Example 1
::例1

Graph the exponential function $\begin{aligned} y = 8 \cdot 3^{- x} \end{aligned}$
::绘制指数函数 y= 8= 83- x 的图形

a.) Here is our table of values and the graph of the function.
::a) 这是我们的数值表和函数图。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = 8 \cdot 3^{- x} \end{aligned}$
-3	$\begin{aligned} y = 8 \cdot 3^{- (- 3)} = 8 \cdot 3^{3} = 216 \end{aligned}$
-2	$\begin{aligned} y = 8 \cdot 3^{- (- 2)} = 8 \cdot 3^{2} = 72 \end{aligned}$
-1	$\begin{aligned} y = 8 \cdot 3^{- (- 1)} = 8 \cdot 3^{1} = 24 \end{aligned}$
0	$\begin{aligned} y = 8 \cdot 3^{0} = 8 \end{aligned}$
1	$\begin{aligned} y = 8 \cdot 3^{- 1} = \frac{8}{3} \end{aligned}$
2	$\begin{aligned} y = 8 \cdot 3^{- 2} = \frac{8}{9} \end{aligned}$

Example 2
::例2

Graph the functions $\begin{aligned} y = 4^{x} \end{aligned}$ and $\begin{aligned} y = 4^{- x} \end{aligned}$ on the same coordinate axes.
::相同坐标轴上的 y= 4x 和 y= 4-x 函数图。

Here is the table of values for the two functions. Looking at the values in the table, we can see that the two functions are “backwards” of each other, in the sense that the values for the two functions are reciprocals.
::这是两个函数的值表。查看表中的值, 我们可以看到这两个函数是“ 后退的 ” , 因为这两个函数的值是对等的。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = 4^{x} \end{aligned}$	$\begin{aligned} y = 4^{- x} \end{aligned}$
-3	$\begin{aligned} y = 4^{- 3} = \frac{1}{64} \end{aligned}$	$\begin{aligned} y = 4^{- (- 3)} = 64 \end{aligned}$
-2	$\begin{aligned} y = 4^{- 2} = \frac{1}{16} \end{aligned}$	$\begin{aligned} y = 4^{- (- 2)} = 16 \end{aligned}$
-1	$\begin{aligned} y = 4^{- 1} = \frac{1}{4} \end{aligned}$	$\begin{aligned} y = 4^{- (- 1)} = 4 \end{aligned}$
0	$\begin{aligned} y = 4^{0} = 1 \end{aligned}$	$\begin{aligned} y = 4^{0} = 1 \end{aligned}$
1	$\begin{aligned} y = 4^{1} = 4 \end{aligned}$	$\begin{aligned} y = 4^{- 1} = \frac{1}{4} \end{aligned}$
2	$\begin{aligned} y = 4^{2} = 16 \end{aligned}$	$\begin{aligned} y = 4^{- 2} = \frac{1}{16} \end{aligned}$
3	$\begin{aligned} y = 4^{3} = 64 \end{aligned}$	$\begin{aligned} y = 4^{- 3} = \frac{1}{64} \end{aligned}$

Here is the graph of the two functions. Notice that the two functions are mirror images of each other if the mirror is placed vertically on the $\begin{aligned} y - \end{aligned}$ axis.
::这是两个函数的图形。请注意, 如果两个函数的镜像垂直放在 y- 轴上, 则这两个函数是彼此的镜像图像。

In the next lesson, you’ll see how functions can be used to represent situations in the real world.
::在下一个教训中, 你会看到如何使用功能来代表真实世界中的局势。

Review
::回顾

Graph the following exponential functions by making a table of values.
::绘制一个数值表,绘制以下指数函数。

$\begin{aligned} y = 3^{x} \end{aligned}$
::y=3x y=3x

$\begin{aligned} y = 5 \cdot 3^{x} \end{aligned}$
::y=5=3x y=5=3x

$\begin{aligned} y = 40 \cdot 4^{x} \end{aligned}$
::y=40=4x

$\begin{aligned} y = 3 \cdot 10^{x} \end{aligned}$
::y=310x

Graph the following exponential functions.
::图形显示以下指数函数。

$\begin{aligned} y = {(\frac{1}{5})}^{x} \end{aligned}$
::y=( 15x) y=( 15x)

$\begin{aligned} y = 4 \cdot {(\frac{2}{3})}^{x} \end{aligned}$
::y=4( 23) x

$\begin{aligned} y = 3^{- x} \end{aligned}$
::y=3 - x y=3 - x

$\begin{aligned} y = \frac{3}{4} \cdot 6^{- x} \end{aligned}$
::y=346-x

Which two of the eight graphs above are mirror images of each other?
::上面的八张图中,哪两张图是彼此的镜像图像?

What function would produce a graph that is the mirror image of the one in problem 4?
::哪个函数能产生一个图形, 即问题4中的镜像图像 ?

How else might you write the exponential function in problem 5?
::否则您如何在问题5中写出指数函数 ?

How else might you write the function in problem 6?
::否则您如何在问题 6 中写入函数 ?

Solve the following problems.
::解决以下问题。

A chain letter is sent out to 10 people telling everyone to make 10 copies of the letter and send each one to a new person.
1. Assume that everyone who receives the letter sends it to ten new people and that each cycle takes a week. How many people receive the letter on the sixth week?
  ::假设每个收到这封信的人都寄给10个新人,每个周期需要一周。第六周有多少人收到这封信?
2. What if everyone only sends the letter to 9 new people? How many people will then get letters on the sixth week?
  ::如果每个人都把信寄给9个新人呢?
::向10个人发送了一条链条信件,要求每个人将信复制10份,并每封寄给一个新的人。假设每个收到信的人将信寄给10个新人,每个周期需要一周。在第六周,有多少人收到信?如果每个人都将信寄给9个新人,会怎样?然后,在第六周,会有多少人收到信?

Nadia received $200 for her $\begin{aligned} 10^{t h} \end{aligned}$ birthday. If she saves it in a bank account with 7.5% interest compounded yearly, how much money will she have in the bank by her $\begin{aligned} 21^{s t} \end{aligned}$ birthday?
::Nadia10岁生日时收到200美元,如果她每年以7.5%的利息在银行账户中存款,那么她到21岁生日时在银行会有多少钱?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

指数函数图图

Graphs of Exponential Functions ::指数函数图图

Exponential Functions ::指数函数

Compare Graphs of Exponential Functions ::比较指数函数图

Comparing Graphs ::比较图表

Example C ::例例C

Graphing Multiple Functions ::绘制多个函数

Example E ::例E

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)